This is related to my previous question, which I have managed to re-formulate but still cannot solve. It's a physics/mechanics problem I'm working on. It may well be the case that I am not Googling the right terms to find the solution, so please correct me if this is the case.
My function is implicitly defined as follows: $$F(r,z,b) = \frac{3r^2 - a}{(r^2 + z^2)^{3/2}} + \frac{3az^2}{(r^2+z^2)^{5/2}} - b = 0$$ where $a$ is a real constant greater than zero, $b>0$ and $a<r^2$.
I first need to find gradient of $z(r)$ with respect to $b$, and then integrate the resulting gradient function with respect to $r$. i.e.: $\int \frac{dz}{db}dr$.
I can find the gradient $dz/db$ easily enough using implicit differentiation: $$\frac{dz}{db} = -\frac{\left(r^2+z^2\right)^{7/2}}{a \left(6 z^3-9 r^2 z\right)+9 r^2 z \left(r^2+z^2\right)}$$
But how I can easily integrate this w.r.t. $r$, given that $z$ is itself a function of $r$ (in that they are both defined in the original implicit equation)?
Ultimately I need a result that depends only on $a$ and $r$ such that I can compute the integral between limits (i.e. $z$ needs to disappear).